Interpretations of probability

State whether the following statements are true or false, justifying your answer:

1. a) "I have rolled this fair die $20$ consecutive times without getting a $5$. According to the law of large numbers, on the next roll it is very likely that a $5$ will finally come up."
2. b) "Even though no $5$ has come up in the last $20$ rolls, the probability of getting one on the next roll remains $\frac{1}{6}$: chance has no memory."

The repeated tossing of a coin has been simulated on a computer and the relative frequency of HEADS has been recorded after $10$, $100$, $1.000$ and $10.000$ tosses, as shown in the graph.

1. a) Towards what value does the relative frequency seem to stabilise? What probability would you estimate for HEADS?
2. b) What law justifies this behaviour? State it in your own words.
3. c) Explain why the estimate with $10$ tosses is unreliable even if the experiment is correct.

After simulating $10.000$ coin flips on a computer, the relative frequency of HEADS turned out to be $0,5003$. Two students comment on the result:

• Student A: "The coin is biased, because if it were fair the relative frequency would be exactly $0,5$."

• Student B: "The exact probability of HEADS for this coin is $0,5003$."

1. a) Analyse the error in each of the two arguments.
2. b) Write down a correct conclusion about what the experiment allows us to state regarding the coin.

A drawing pin is dropped $400$ times and lands point up on $260$ occasions.

1. a) Estimate the probability that the drawing pin lands point up.
2. b) Explain why Laplace's rule cannot be applied in this experiment.
3. c) Which interpretation of probability did you use in part a)?

The weather app announces: "tomorrow, $70\ \%$ probability of rain".

1. a) Explain what this statement means: is it a classical (Laplace), frequentist or subjective probability? What data do you think it is based on?
2. b) Three friends assign probabilities to Saturday's match. The first says: "my team wins with probability $0,5$, draws with $0,3$ and loses with $0,3$". The second says: "it wins with $0,6$, draws with $0,3$ and loses with $0,1$". Are both assignments coherent? Why?
3. c) Can a subjective probability be "incorrect" even if it is coherent? Comment briefly.

Three people are discussing the probability that the 8:00 bus arrives late:

• Person 1: "Out of the last $200$ school days, it arrived late $24$ times, so the probability is $0,12$."

• Person 2: "The bus arrives either on time or late: two possible outcomes, so the probability of arriving late is $\frac{1}{2}$."

• Person 3: "I'd say there is a $40\ \%$ probability of it arriving late; that's what my experience tells me."

1. a) Identify which interpretation of probability (frequentist, classical or Laplace, and subjective) corresponds to each statement.
2. b) The reasoning of one of the people is incorrect. Which one is it and why?
3. c) What data would you collect to improve each of the three estimates?

At a fairground stall, a game called "the lucky disc" is played: a disc of $2\ \text{cm}$ in diameter is dropped onto a grid of squares of side length $5\ \text{cm}$, as shown in the figure, and you win if the disc lands without touching any line.

1. a) Calculate the theoretical probability of winning. (Hint: the position of the disc is determined by its centre; reason about which region of each square the centre must land in.)
2. b) A player drops the disc $300$ times and wins $114$ times. Calculate the relative frequency of winning and compare it with the theoretical probability.
3. c) Does the experimental result contradict the theoretical probability? Give reasons for your answer.

A dart is thrown at a circular target made of three concentric circles of radii $r$, $2r$ and $3r$. Assuming that the dart always hits the target and that the probability of hitting a region is directly proportional to its area, calculate the probability that the dart hits the middle annulus (the region contained between the circle of radius $r$ and that of radius $2r$).